# What's the meaning of $C$-embedded?

What's the meaning of $C$-embedded? It is a topological notion. Thanks ahead.

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–  Micah Sep 2 '12 at 7:15
thanks for the link. it is useful for me. –  Paul Sep 2 '12 at 7:21
A set $A \subset X$ ($X$ is a topological space) is $C$-embedded in $X$ iff every real-valued continuous function $f$ defined on $A$ has a continuous extension $g$ from $X$ to $\mathbb{R}$ (so $g(x) = f(x)$ for all $x \in A$).
A related notion of $C^{\ast}$-embedded exist where continuous real-valued functions are (in both cases) replaced by bounded real-valued continuous functions.
The Tietze theorem basically says that a closed subset $A$ of a normal space $X$ is $C$ and $C^{\ast}$-embedded in it.